Arvosiset

Arvosiset is a term used in Finnish mathematical literature to denote the image set or range of a valuation or measurable function applied to a given domain. The concept is used to discuss the scope of values that a function can take, relative to its codomain and the elements of its domain.

Formally, if f is a function from a set X to a codomain Y, the arvosiset of f is the set f(X) which consists of all values that f attains on X. This notion helps describe the extent of variation produced by a function and is closely related to the standard idea of the image or range in English-language texts. In probability and statistics, if a random variable X maps outcomes to a set of values, the arvosiset of X corresponds to the image of X, i.e., the values that X can possibly take.

Examples illustrate the concept: for f(n) = n mod 3 with domain X = natural numbers, the arvosiset is {0, 1, 2}. If a random variable takes only the values 0 and 1, its arvosiset is {0, 1}.

Properties include that the arvosiset is always a subset of the codomain and is determined entirely by the domain and the function. It need not be a single interval; for many functions the arvosiset can be a union of disjoint values or intervals, especially in the case of piecewise or discrete mappings.

In practice, the term is most common in Finnish-language texts. In English, one would typically refer to the image, range, or set of values attained by the function. Etymologically, arvosiset derives from arvo (value) and setti (set). See also image, range, valuation, measure theory.